Celebratio Mathematica — Chung

[1]K. L. Chung: “Contributions to the theory of Markov chains,” J. Research Nat. Bur. Standards 50 : 4 (April 1953), pp. 203–208. Research Paper 2411. MR 0055608 Zbl 0053.27201 article

The fundamentals of the theory of denumerable Markov chains with stationary transition probabilities were laid down by Kolmogorov and further work was done by Doblin. The theory of recurrent events of Feller is closely related, if not coextensive. Some new results obtained by T. E. Harris turn out to tie up very nicely with some amplifications of Doblin’s work. Harris was led to consider the probabilities of hitting one state before another, starting from a third one. This idea of considering three states, one initial, one “taboo”, and one final, is more fully developed in the present work. The notion of first passage time of the “union” or “intersection” of two states is also introduced here. The interplay between these notions is illustrated.

[2]K. L. Chung: “Contributions to the theory of Markov chains, II,” Trans. Amer. Math. Soc. 76 (1954), pp. 397–419. MR 0063603 Zbl 0058.34602 article

[3]K. L. Chung: “Foundations of the theory of continuous parameter Markov chains,” pp. 29–40 in Proceedings of the third Berkeley symposium on mathematical statistics and probability, 1954–1955, vol. II: Contributions to probability theory. Edited by J. Neyman. University of California Press (Berkeley, CA and Los Angeles, CA), 1956. MR 0084884 Zbl 0071.35001 inproceedings

This paper is exclusively concerned with continuous parameter Markov processes with a denumerably infinite number of states and stationary transition matrix function. The foundations of the proper theory of such processes, as distinguished from that of the discrete parameter version, or of Markov processes which are either more special (for example, general state space; nonstationary transition matrix function), were laid by Doob [1942; 1945], and Lévy [1951; 1952; 1953]. Roughly speaking it was Lévy in 1952 who drew, in his inimitable way, the comprehensive picture while Doob, ten years earlier, had supplied the essential ingredients. The present effor aims at a synthesis of the most fundamental parts of the theory, made possible by the contributions of these two authors. While the results given here generally extend and clarify those in the cited literature, immense credit must go to Professors Lévy and Doob for the inspiration of their pioneer work. To them I am also indebted for much valuable discussion through correspondence and conversation. An attempt is made in the presentation to be quite formal and rigorous, in the spirit of Doob’s already classic treatise [1953]. Further developments of the theory will be published elsewhere.

[4]K. L. Chung: “Some new developments in Markov chains,” Trans. Amer. Math. Soc. 81 (1956), pp. 195–210. MR 0075481 Zbl 0075.14001 article

The purpose of this paper is to point out some new connections between the sample function behavior and the analytical properties of the transition probability functions of a continuous parameter Markov chain with stationary transition probabilities. The main idea is that of a post-exit process derived from the original process by considering its evolution after the exit from a stable state. This leads to various relations between conditional probabilities all of which are “intuitively obvious” but require sometimes painstaking proofs if probability theory like any other branch of mathematics is to be treated as a discipline in logic. These relations imply certain analytical properties of the transition functions, obtained recently by D. G. Austin by purely analytical means, and exhibit them in connection with other quantities introduced in this paper. They also complete some results due to Doob and Lévy. The well-known differential equations of Kolmogorov are seen to be limiting cases of certain more generally valid differential equations involving a continuous parameter. One of these expresses the fundamental transition property of the post-exit process, and the other a similar property of the renewal density of an imbedded renewal process.

[5]K. L. Chung: “Some aspects of continuous parameter Markov chains,” Publ. Inst. Statist. Univ. Paris 6 (1957), pp. 271–287. MR 0102128 Zbl 0082.34502 article

[6]K. L. Chung: “On a basic property of Markov chains,” Ann. of Math. (2) 68 : 1 (July 1958), pp. 126–149. MR 0100924 Zbl 0228.60023 article

[7]K. L. Chung: “Some remarks on taboo probabilities,” Illinois J. Math. 5 : 3 (1961), pp. 431–435. MR 0131902 Zbl 0101.11601 article

[8]K. L. Chung: “Probabilistic methods in Markov chains,” pp. 35–56 in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, vol. II. Edited by J. Neyman. University of California Press (Berkeley, CA), 1961. MR 0131901 Zbl 0106.33303 incollection

[9]K. L. Chung: “On the Martin boundary for Markov chains,” Proc. Nat. Acad. Sci. U.S.A. 48 : 6 (June 1962), pp. 963–968. MR 0145584 Zbl 0228.60030 article

[10]K. L. Chung: “On the boundary theory for Markov chains,” Acta Math. 110 (1963), pp. 19–77. MR 0150829 Zbl 0292.60121 article

[11]K. L. Chung: “On the boundary theory for Markov chains, II,” Acta Math. 115 : 1 (1966), pp. 111–163. MR 0189117 Zbl 0315.60041 article

[12]K. L. Chung: “Boundary behavior of Markov chains and its contributions to general processes,” pp. 499–505 in Actes du Congrès International des Mathématiciens (Nice, France, September 1–10, 1970), Tome 2. Gauthier-Villars (Paris), 1971. MR 0420882 Zbl 0298.60047 incollection

[13]K. L. Chung: “‘Markov chain must have a beginning.’ In memory of Prof. Loo Keng Hua,” J. Math. Res. Expo. 1 (1986), pp. 1–3. MR 858088 Zbl 0632.60064 article